THE DOUBTER
product of (rational) primes in only one way. From this theorem
springs all the intricate theory of divisibility for rational
integers. Unfortunately the fundamental theorem does not hold
in all algebraic number fields of degree greater than one, and
the result is chaos.
To give an instance (it is the stock example usually exhibited
in text-books on the subject), in the field F[ V — 5] we have
6 = 2X3 = (1 + V~5) X (1 - V~5);
each of 2, 3,1 -f V - 5,1 - V~^5 is a prime in this field (as
may be verified with some ingenuity), so that 6, in this field, is
not uniquely decomposable into a product of primes.
It may be stated here that Kronecker overcame this difficulty
by a beautiful method which is too detailed to be explained
untechnically, and that Dedekind did likewise by a totally
different method which is much easier to grasp, and which will
be noted when we consider his life. Dedekind's method is the
one in widest use to-day, but this does not imply that Kro-
necker's is less powerful, nor that it will not come into favour
when more arithmeticians become familiar with it.
In his dissertation of 1845 Kronecker attacked the theory of
the units in certain special fields - those defined by the equa-
tions arising from the algebraic formulation of Gauss' problem
to divide the circumference of a circle into n equal parts or,
what is the same, to construct a regular polygon of n sides,
We can now close up one part of the account opened by
Fennat. In struggling to prove Fennat's "Last Theorem* that
x" -f yn = zn is impossible in rational integers x, y, z (none zero)
if n is an integer greater than 2, arithmeticians took what looks
like a natural step and resolved the left-hand side, a?n -f- yn,
into its n factors of the first degree (as is done in the usual
second course of school algebra). This led to the exhaustive
investigation of the algebraic number field mentioned above in
connexion with Gauss' problem - after serious but readily
understandable mistakes had been made.
The problem at first was studded with pitfalls, into which
many a competent mathematician and at least one great one -
Cauchy - tumbled headlong. Cauchy assumed as a matter of
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